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DFG-GK1100 Kolleg SeminarUniversität Ulm – June 4th, 2012Coherent Modeling of the Risk in Mortality Projections:Theory and ApplicationsNan Zhu and Daniel BauerDepartment of RMI, Georgia State University

Page 2 / 34Introduction1IntroductionMotivationLiterature Review: MortalityLiterature Review: Term Structure Models2Factor Analysis of Mortality Forecasts3Forward Mortality Factor Models4A Non-negative Model Variant5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 3 / 34 IntroductionMotivationConsider a 60 year old individual who faces systematic mortality risk:1. Currently expected to die with probability of 0.7% next year (mortality rate).How uncertain is this appraisal? What are the chances that the rate is 0.65%or 0.75%? Risk in mortality rates2. Currently expected to live 25.1 more years. How uncertain is this number?What are the chances that it changes to 23.7 or 24.5 next year? Risk in mortality projections While the two questions are related, and the distinction does not matter intheory, it is relevant for the econometrical/statistical approach For personal financial planning/household finance, insurers’ liability riskevaluation, and government/public economics, question 2 may be moresuitableDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 3 / 34 IntroductionMotivationConsider a 60 year old individual who faces systematic mortality risk:1. Currently expected to die with probability of 0.7% next year (mortality rate).How uncertain is this appraisal? What are the chances that the rate is 0.65%or 0.75%? Risk in mortality rates2. Currently expected to live 25.1 more years. How uncertain is this number?What are the chances that it changes to 23.7 or 24.5 next year? Risk in mortality projections While the two questions are related, and the distinction does not matter intheory, it is relevant for the econometrical/statistical approachFor personal financial planning/household finance, insurers’ liability riskevaluation, and government/public economics, question 2 may be moresuitableDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 3 / 34 IntroductionMotivationConsider a 60 year old individual who faces systematic mortality risk:1. Currently expected to die with probability of 0.7% next year (mortality rate).How uncertain is this appraisal? What are the chances that the rate is 0.65%or 0.75%? Risk in mortality rates2. Currently expected to live 25.1 more years. How uncertain is this number?What are the chances that it changes to 23.7 or 24.5 next year? Risk in mortality projections While the two questions are related, and the distinction does not matter intheory, it is relevant for the econometrical/statistical approachFor personal financial planning/household finance, insurers’ liability riskevaluation, and government/public economics, question 2 may be moresuitableDaniel BauerCoherent Modeling of the Risk in Mortality Projections

IntroductionMotivationPage 4 / 34!!!!!!!!!!!!!!!!!!!!"!!"!!"!!!!!!!!!!!!!!!!!!"# %! !!!!! &!!!'!!!!!!!!!(! !!!!!!!!!!()'!Current stochastic mortality models focus on stochastically forecastingmortality rates (question 1) red in graphic This essay considers the risk in mortality projections (question 2) blue in graphicDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 5 / 34 IntroductionComparison with Interest Rate Models/Yiled Curve ModelsAnalogy in the motivation:I Want to forecast yield curve p(T 1, τ ) based on p(t, τ ), 0 t T (similarto question 2)R t τI Know p(t, τ ) EQrs ds)], so (theoretically) modeling risk in rtt [exp( tshould be sufficient (similar to question 1)I But to identify reasonable specification, we also need to considercross-sectional data (persistence vs. transience of errors, etc.) But there are key differences:I Age: additional dimension?How does age enter into model equation?What are appropriate models?I Data?Age/term panelsWhere do we get "mortality forecasts"?I Risk adjustments (P vs. Q) and consequencesDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 5 / 34 IntroductionComparison with Interest Rate Models/Yiled Curve ModelsAnalogy in the motivation:I Want to forecast yield curve p(T 1, τ ) based on p(t, τ ), 0 t T (similarto question 2)R t τI Know p(t, τ ) EQrs ds)], so (theoretically) modeling risk in rtt [exp( tshould be sufficient (similar to question 1)I But to identify reasonable specification, we also need to considercross-sectional data (persistence vs. transience of errors, etc.) But there are key differences:I Age: additional dimension?How does age enter into model equation?What are appropriate models?I Data?Age/term panelsWhere do we get "mortality forecasts"?I Risk adjustments (P vs. Q) and consequencesDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 6 / 34IntroductionLiterature Review: MortalityLarge literature with various methods on mortality forecasting: Lee-Carter approach (Lee and Carter (JASA, 1992)):(1)(2) (2)log m(t, x) βx βx κt CBD-Perks model (Cairns et al. (JRI, 2006)):(1)logit q(t, x) κt (2) κ2 (x x̄)P-splines method (Currie et al. (Statistical Modeling, 2004)):non-parametric modelAll these methodologies. rely on past mortality data to project the mortality experience in someoptimal sense. pay little attention on the uncertainty (error estimates) associated with theprojections. may fail to identify the transiency of different random sourcesDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 6 / 34IntroductionLiterature Review: MortalityLarge literature with various methods on mortality forecasting: Lee-Carter approach (Lee and Carter (JASA, 1992)):(1)(2) (2)log m(t, x) βx βx κt CBD-Perks model (Cairns et al. (JRI, 2006)):(1)logit q(t, x) κt (2) κ2 (x x̄)P-splines method (Currie et al. (Statistical Modeling, 2004)):non-parametric modelAll these methodologies. rely on past mortality data to project the mortality experience in someoptimal sense. pay little attention on the uncertainty (error estimates) associated with theprojections. may fail to identify the transiency of different random sourcesDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 7 / 34 IntroductionLiterature Review: Term Structure ModelsFactor modelsI Factor analysis: Litterman and Scheinkman (Journal of Fixed Income,1991), Rebonato (Interest Rate Option Models, 1998).I VaR models: Fama and Bliss (AER, 1987), Diebold and Li (JEconometrics,2006), Duffee (JFin, 2002). Forward rate modelsI HJM model: Heath et al. (Econometrica, 1992), Filipović (Lecture Notes inMathematics, 2004).I In the case of mortality: Cairns et al. (ASTIN Bull., 2006), Barbarin (IME,2008), Bauer et al. (IME, 2010), Bauer et al. (2011). Finite-dimensional realizations/Affine term structure modelsI FDR: Björk and Gombani (FinStoch, 1999), Björk and Svensson(MathFinanc, 2001).I Affine Models: Duffie and Kan (MathFinanc, 1996), Duffie et al.(Econometrica, 2000), Piazzesi (Handbook of Financial Econometrics,2010). Improvement of Cross-sectional constraintsI Joslin et al. (RFS, 2011), Duffee (2011).Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 8 / 34 IntroductionPreview of ResultsConfidence intervals for life expectancies in one year for a now 20 yearold female (USA)Comparison with conventional mortality forecasting approach –Lee-CarterI Generally underestimate the risk in mortality projectionsDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 9 / 34Factor Analysis of Mortality Forecasts1Introduction2Factor Analysis of Mortality ForecastsForward Force of Mortality FrameworkData and Projection MethodsFactor AnalysisSimple Factor Models3Forward Mortality Factor Models4A Non-negative Model Variant5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 10 / 34Factor Analysis of Mortality ForecastsForward Force of Mortality FrameworkForward survival probabilities: {τ px (t) (τ, x) C}Forward force of mortality (easier to model/work with):µt (τ, x) log{τ px (t)} τConsider time-homogeneous, Gaussian modelsdµt (A µt Λ) dt Σ dWt x) (infinitesimal generator of a strongly continuousA ( τsemigroup)Wt : d-dimensional Brownian motionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 11 / 34Factor Analysis of Mortality ForecastsForward Force of Mortality FrameworkDefineFl (tj , tj 1 , (τ, x)) log τ l t log τ l px (tj 1 )pj 1 tj x tj 1 t (tj )jτ px (tj 1 )τ tj 1 tj px tj 1 tj (tj )τ l px (tj 1 )τ px (tj 1 ) τ l tj 1 tj px tj 1 tj (tj ) τ tj 1 tj px tj 1 tj (tj ) l: time lag, tj 1 tj Measures the log change of the l-year marginal survival probability!!!!!!!"# ############!" ! "# ##!" ! " %####!!!!!!!" "## #########!" ! "# ################!#######!" ! " %!!!!!!!!Daniel Bauer!!!!!!!!"# ########%!Coherent Modeling of the Risk in Mortality Projections

Page 11 / 34Factor Analysis of Mortality ForecastsForward Force of Mortality FrameworkDefineFl (tj , tj 1 , (τ, x)) log τ l t log τ l px (tj 1 )pj 1 tj x tj 1 t (tj )jτ px (tj 1 )τ tj 1 tj px tj 1 tj (tj )τ l px (tj 1 )τ px (tj 1 ) τ l tj 1 tj px tj 1 tj (tj ) τ tj 1 tj px tj 1 tj (tj ) l: time lag, tj 1 tj Measures the log change of the l-year marginal survival probability!!!!!!!"# ############!" ! "# ##!" ! " %####!!!!!!!" "## #########!" ! "# ################!#######!" ! " %!!!!!!!!Daniel Bauer!!!!!!!!"# ########%!Coherent Modeling of the Risk in Mortality Projections

Page 12 / 34Factor Analysis of Mortality ForecastsForward Force of Mortality FrameworkProposition IIf tj 1 tj , the vectorsF̄l (tj , tj 1 )Fl (tj , tj 1 , (τ, x))ptj 1 tj! ω(τ, x) Fl (tj , tj 1 , (τ1 , x1 ))Fl (tj , tj 1 , (τK , xK ))ppω(τ1 , x1 ) , . . . , ω(τK , xK ) tj 1 tjtj 1 tj(τ,x) C̃j 1, 2, . . . , N 1 are i.i.d. Gaussian distributed.(the weights ω(τ, x) allow for different weighting of future – e.g. p(t, τ ))Daniel BauerCoherent Modeling of the Risk in Mortality Projections!,

Page 13 / 34Factor Analysis of Mortality ForecastsData and Projection Methods"True" Mortality forecasts from market place or insurance prices notavailable or at least not abundant/noisy.Raw data: (deterministic) mortality forecasts generated from rolling windowsof past mortality experience Regions: England/Wales (ENW), France (FRA), Japan (JPN), UnitedStates (USA), and West Germany (FRG) Genders: male and female Years: 1956-2006 22 projections (each using mortality experience of past 30 years) Methods:Lee-CarterI Weighted-least-squares algorithmI Ages: 0-95CBD-PerksI Basic model w/o cohort effectI Ages: 25-95P-splinesI Fixing the degree of freedom (df ) at 20I Ages: 25-95Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 13 / 34Factor Analysis of Mortality ForecastsData and Projection Methods"True" Mortality forecasts from market place or insurance prices notavailable or at least not abundant/noisy.Raw data: (deterministic) mortality forecasts generated from rolling windowsof past mortality experience Regions: England/Wales (ENW), France (FRA), Japan (JPN), UnitedStates (USA), and West Germany (FRG) Genders: male and female Years: 1956-2006 22 projections (each using mortality experience of past 30 years) Methods:Lee-CarterI Weighted-least-squares algorithmI Ages: 0-95CBD-PerksI Basic model w/o cohort effectI Ages: 25-95P-splinesI Fixing the degree of freedom (df ) at 20I Ages: 25-95Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Factor Analysis of Mortality ForecastsFactor AnalysisPage 14 / 34With tj 1 tj 1, F̄l (tj , tj 1 ) are i.i.d. Gaussian F̄l (tj , tj 1 ) a bZj j a RK , b RK d , factors Zj Rd with E(Zj ) 0 and Cov(Zj ) Id dEstimates of a, b, and the number of factors, d, from principal componentanalysis: Decompose empirical covariance matrix of F̄l (tj , tj 1 ): Σ̂Σ̂ U diag{λ1 , . . . , λK } U 0 KXλν uν uν0ν 1 dXλν uν uν0 Covν 1 !uνpλν Zν,jν 1Determinethe value of dPI dXdν 1PKν 1λνλν ξInvestigate the shape of the factorsDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Factor Analysis of Mortality ForecastsFactor AnalysisPage 14 / 34With tj 1 tj 1, F̄l (tj , tj 1 ) are i.i.d. Gaussian F̄l (tj , tj 1 ) a bZj j a RK , b RK d , factors Zj Rd with E(Zj ) 0 and Cov(Zj ) Id dEstimates of a, b, and the number of factors, d, from principal componentanalysis: Decompose empirical covariance matrix of F̄l (tj , tj 1 ): Σ̂Σ̂ U diag{λ1 , . . . , λK } U 0 KXλν uν uν0ν 1 dXλν uν uν0 Covν 1 !uνpλν Zν,jν 1Determinethe value of dPI dXdν 1PKν 1λνλν ξInvestigate the shape of the factorsDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 15 / 34Factor Analysis of Mortality ForecastsPCA Results: The Lee-Carter ApproachCountryFactorFemale PopulationValuePercentageMale PopulationValuePercentageUnited Statesλ1λ2λ3λ4λ5λ6 4.50 10 21.80 10 36.15 10 44.79 10 42.29 10 42.07 10 493.13%3.70%1.27%-7.51 10 28.30 10 32.60 10 31.91 10 38.57 10 44.22 10 484.29%9.26%2.91%-First factor (PC1) dominates the restHigher volatility for male populationI Higher absolute valueI Lower weight of first principle componentDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 16 / 34Factor Analysis of Mortality ForecastsSurfaces of Eigenvectors: ENW/Lee-CarterDaniel Bauer(a) female, PC1(b) female, PC2(c) male, PC1(d) male, PC2Coherent Modeling of the Risk in Mortality Projections

Factor Analysis of Mortality ForecastsSummaryPage 17 / 34 Very similar shapes exhibited across differentcountries/genders/forecasting methods (at least for the first factor) PC1I Systematic, increasing in age/termI Forward forces of mortality for high ages in the far future more volatile thanin the near future slope factor PC2I Male (consistently over time decreasing influence, even generate inverserelationship)I Female (sometimes similar, sometimes rather unsystematic) twist factor d 1 for female population; d 2 for male population (Lee-Carter)d 1 for both genders (CBD & P-spline)Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Factor Analysis of Mortality ForecastsSummaryPage 17 / 34 Very similar shapes exhibited across differentcountries/genders/forecasting methods (at least for the first factor) PC1I Systematic, increasing in age/termI Forward forces of mortality for high ages in the far future more volatile thanin the near future slope factor PC2I Male (consistently over time decreasing influence, even generate inverserelationship)I Female (sometimes similar, sometimes rather unsystematic) twist factor d 1 for female population; d 2 for male population (Lee-Carter)d 1 for both genders (CBD & P-spline)Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Factor Analysis of Mortality ForecastsSummaryPage 17 / 34 Very similar shapes exhibited across differentcountries/genders/forecasting methods (at least for the first factor) PC1I Systematic, increasing in age/termI Forward forces of mortality for high ages in the far future more volatile thanin the near future slope factor PC2I Male (consistently over time decreasing influence, even generate inverserelationship)I Female (sometimes similar, sometimes rather unsystematic) twist factor d 1 for female population; d 2 for male population (Lee-Carter)d 1 for both genders (CBD & P-spline)Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 18 / 34Factor Analysis of Mortality ForecastsSimple Factor ModelsSimple factor models from the factor analysis 4Yν (j) (uν λν )T F̄l (uν λν )T E[F̄l ] λν Zν,j , ν 1, . . . , d Regression equation of F̄l (tj , tj 1 ) on Yν (j) (similar to Diebold and Li,(JEconometrics, 2006)):F̄l (tj , tj 1 ) 4 d Xuν [Yν (j) (uν λν )T E[F̄l (tj , tj 1 )]] jE F̄l (tj , tj 1 ) λνν 1m̃ dXs̃ν Yν (j) jν 1 Simple, easy-to-estimate mortality forecasting methodologiesI Simulate Yν (N) N(µsY ,ν , σYs ,ν ) with (µsY ,ν , σYs ,ν ) as sample mean andstandard error of Yν (j), j 1, . . . , N 1PI Forecast: F̄l (tN , tN 1 ) m̃ dν 1 s̃ν Yν (N) together w/ known τ px (tN )Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 18 / 34Factor Analysis of Mortality ForecastsSimple Factor ModelsSimple factor models from the factor analysis 4Yν (j) (uν λν )T F̄l (uν λν )T E[F̄l ] λν Zν,j , ν 1, . . . , d Regression equation of F̄l (tj , tj 1 ) on Yν (j) (similar to Diebold and Li,(JEconometrics, 2006)):F̄l (tj , tj 1 ) 4 d Xuν [Yν (j) (uν λν )T E[F̄l (tj , tj 1 )]] jE F̄l (tj , tj 1 ) λνν 1m̃ dXs̃ν Yν (j) jν 1 Simple, easy-to-estimate mortality forecasting methodologiesI Simulate Yν (N) N(µsY ,ν , σYs ,ν ) with (µsY ,ν , σYs ,ν ) as sample mean andstandard error of Yν (j), j 1, . . . , N 1PI Forecast: F̄l (tN , tN 1 ) m̃ dν 1 s̃ν Yν (N) together w/ known τ px (tN )Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 19 / 34Forward Mortality Factor Models1Introduction2Factor Analysis of Mortality Forecasts3Forward Mortality Factor ModelsSelf-consistency ConditionMaximum Likelihood EstimationShould the Self-consistency Condition be Imposed?4A Non-negative Model Variant5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Forward Mortality Factor ModelsSelf-consistency Conditiono n RtMartingale property of exp 0 µs (0, x0 s) dsPage 20 / 34 oihn RTEt exp 0 µs (0, x0 s) dson Rt exp 0 µs (0, x0 s) dsT t px0 t (t) T t px0 t (t) Drift condition (Cor. 3.1 in Bauer et al. (2011)):Z τα(τ, x) σ(τ, x) σ 0 (s, x) ds0 Bauer et al. (2011): µt allows for a Gaussian finite-dimensionalrealization (FDR) iffσ(τ, x) C(x τ ) exp{Mτ } N µt (τ, x) µ0 (τ t, x t) Rtα(τ t s, x t s) dsZ texp {M (t s)} N dWs C(x τ ) exp {M τ } 0{z} 0 Zt (state process)Proposition II: Possible to consider each factor separatelyDaniel BauerCoherent Modeling of the Risk in Mortality Projectionst 0:

Page 21 / 34 Forward Mortality Factor ModelsAnalysis of C(x), M, and NZdFl (τ, x) E[Fl (τ, x)] ττ 1Z eM( s) N dWsC(x v ) eMv dv 0{z} {z} O(τ,x) Z uν λν (Oν (τi , xi ))1 i K exp{Mν (tj 1 tj )}Nν Estimate Cν (x), Mν , and Nν via two-step identification:1. Estimate M and N w/o functional assumptions on C(x)?Rely on examples from interest rate modeling to find convenient shapes, inparticular Björk and Gombani (FinStoch, 1999)2. Functional assumptions for C(x)Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 22 / 34Forward Mortality Factor ModelsThe Slope Factor: U.S. Dataσ1 (τ, x) kexp(c(x τ ) d)(a τ ) exp( bτ )(1 exp(c(x τ ) d))(e) female, Lee-Carter(f) female, CBD(g) female, P-spline(h) male, Lee-Carter(i) male, CBD(j) male, P-splineDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 23 / 34Forward Mortality Factor ModelsThe Twist Factor: Male/Lee-Carter σ2 (τ, x) k1exp(c1 (x τ ) d1 )exp(c2 (x τ ) d2 ) k21 exp(c1 (x τ ) d1 )1 exp(c2 (x τ ) d2 )(k) ENW(l) FRA(n) USADaniel Bauer exp{M2 τ }N2(m) JPN(o) FRGCoherent Modeling of the Risk in Mortality Projections

Page 24 / 34Forward Mortality Factor ModelsMaximum Likelihood Estimation From matching {C, M, N} to u λ we can estimate parameter values, however Only consider the variance part of F̄ (tj , tj 1 ) while neglecting the moment:E(F̄ (tj , tj 1 )) Need to consider the self-consistency condition:Z τα(τ, x) σ(τ, x) σ 0 (s, x) ds0 In addition, allow for non-systematic deviationsI F̄ obs (tj , tj 1 ) F̄ mod (tj , tj 1 ) jI j N(0, α · diag{Σ}), j 1, . . . , N 1I α the weight (in the PCA) of all eigenvalues not considered in our model F̄ obs (tj , tj 1 ) N(µ̄, Σ̃ Σ α · diag{Σ})F Maximum likelihood estimationDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 24 / 34Forward Mortality Factor ModelsMaximum Likelihood Estimation From matching {C, M, N} to u λ we can estimate parameter values, however Only consider the variance part of F̄ (tj , tj 1 ) while neglecting the moment:E(F̄ (tj , tj 1 )) Need to consider the self-consistency condition:Z τα(τ, x) σ(τ, x) σ 0 (s, x) ds0 In addition, allow for non-systematic deviationsI F̄ obs (tj , tj 1 ) F̄ mod (tj , tj 1 ) jI j N(0, α · diag{Σ}), j 1, . . . , N 1I α the weight (in the PCA) of all eigenvalues not considered in our model F̄ obs (tj , tj 1 ) N(µ̄, Σ̃ Σ α · diag{Σ})F Maximum likelihood estimationDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 25 / 34Forward Mortality Factor ModelsShould the Self-consistency Condition be Imposed?Debate on the necessity of imposing cross-sectional constraints: Unlike term structure modeling, increase efficiency of estimates ("P Q")BUT: Models not satisfying cross-sectional constraints should producesimilar forecasts. In particular, imposing cross-sectional constraintsshould not invalidate estimates in their absence (Duffee (2011)) Test self-consistency of forecasting approachesAble to compare estimates for µY and σY in three cases:1. Directly calculated as the sample mean and standard error from the dataset: (µsY , σYs )2. Estimated based on the specific functional assumption on σ(τ, x) butwithout the self-consistency condition in place: (µuY , σYu )3. Estimated via the MLE from the previous subsection under the specificfunctional assumption on σ(τ, x) with the self-consistency condition inplace: (µcY , σYc )Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 25 / 34Forward Mortality Factor ModelsShould the Self-consistency Condition be Imposed?Debate on the necessity of imposing cross-sectional constraints: Unlike term structure modeling, increase efficiency of estimates ("P Q")BUT: Models not satisfying cross-sectional constraints should producesimilar forecasts. In particular, imposing cross-sectional constraintsshould not invalidate estimates in their absence (Duffee (2011)) Test self-consistency of forecasting approachesAble to compare estimates for µY and σY in three cases:1. Directly calculated as the sample mean and standard error from the dataset: (µsY , σYs )2. Estimated based on the specific functional assumption on σ(τ, x) butwithout the self-consistency condition in place: (µuY , σYu )3. Estimated via the MLE from the previous subsection under the specificfunctional assumption on σ(τ, x) with the self-consistency condition inplace: (µcY , σYc )Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Forward Mortality Factor ModelsShould the Self-consistency Condition be Imposed?Page 26 / 157(0.0047, 0.0266)0.0235(0.0184, 0.0347)0.01560.02350.01700.0282CBD-Perks0.0016( 0.0079, 0.0111)0.0204(0.0160, 0.0302)0.00160.02040.00430.1418P-splines 0.0249( 0.1041, 0.0543)0.1698 0.0249(0.1331, 0.2512)0.1698 0.05750.9968 (µuY , σYu ) and (µsY , σYs ) are very close indicates good parametric fit(µcY , σYc ) are only close to the (µsY , σYs ) for the Lee-Carter methodFor the other forecasting approaches, σYc considerably exceeds the upperbound of the corresponding confidence intervalEndorse the use of the Lee-Carter method for producing (deterministic)mortality forecastsDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Forward Mortality Factor ModelsShould the Self-consistency Condition be Imposed?Page 26 / 157(0.0047, 0.0266)0.0235(0.0184, 0.0347)0.01560.02350.01700.0282CBD-Perks0.0016( 0.0079, 0.0111)0.0204(0.0160, 0.0302)0.00160.02040.00430.1418P-splines 0.0249( 0.1041, 0.0543)0.1698 0.0249(0.1331, 0.2512)0.1698 0.05750.9968 (µuY , σYu ) and (µsY , σYs ) are very close indicates good parametric fit(µcY , σYc ) are only close to the (µsY , σYs ) for the Lee-Carter methodFor the other forecasting approaches, σYc considerably exceeds the upperbound of the corresponding confidence intervalEndorse the use of the Lee-Carter method for producing (deterministic)mortality forecastsDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 27 / 34A Non-negative Model Variant1Introduction2Factor Analysis of Mortality Forecasts3Forward Mortality Factor Models4A Non-negative Model VariantFramework5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

A Non-negative Model VariantPage 28 / 34FrameworkBased on our specific one-factor assumption, the spot force of mortality isZµt (0, x)tµ0 (t, x t) Ztµ0 (t, x t) (2)α(t s, x t s) ds f (x) Zt00(2)α(t s, x t s) ds ξ2 f (x) f (x) (Zt ξ2 ) {z }(2)Z̃t(1)d Z̃t dZt(2)Z̃t! 2ξ1 b ξ2 b2 ξ1 b20 2b1 Z̃t(2)(2)since µt (0, x) only depends on Z̃t , we thus change Z̃tprocess (analogy: Vasicek model to CIR model) d Z̃t 2ξ1 b ξ2 b2 ξ1 2b1 b20 Z̃t dt dt 1 abadWtto a square-root q(2)(1 ab) Z̃t dWtq(2)a Z̃tPreserves affine structure Calibration w/ generalized Kalman filterDaniel Bauer Coherent Modeling of the Risk in Mortality Projections

Page 29 / 34Application1Introduction2Factor Analysis of Mortality Forecasts3Forward Mortality Factor Models4A Non-negative Model Variant5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 30 / 34ApplicationConfidence Intervals for Future LE: USA Female (After 1 yr)(p) age 20(q) age 30(r) age 40(s) age 50(t) age 60(u) age 70Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 31 / 34ApplicationConfidence Intervals for Future LE: USA Female (After 1 yr)(v) age 20(x) age 50Daniel Bauer(w) age 40(y) age 70Coherent Modeling of the Risk in Mortality Projections

Page 32 / 34Conclusion1Introduction2Factor Analysis of Mortality Forecasts3Forward Mortality Factor Models4A Non-negative Model Variant5Application6ConclusionDaniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 33 / 34ConclusionConclusion Having appropriate estimates for the risk in mortality projections isimportant Common approach may not be suitable to appraise risk within medium orlong-term projections We provide a parsimonious and tractable alternative Mortality Surface Models / Mortality Term Structure Models Applications in the life insurance context: "Applications of ForwardMortality Factor Models in Life Insurance Practice", Geneva Papers,2011, 36: 567-594.Future Work Multiple populationsApplication in household finance: Annuitization decision in portfoliocontext/influence of systematic mortality riskDaniel BauerCoherent Modeling of the Risk in Mortality Projections

ConclusionPage 34 / 34ContactNan Zhu & Daniel Bauernzhu1@student.gsu.edu &dbauer@gsu.eduGeorgia State & www.rmi.gsu.eduThank you!Daniel BauerCoherent Modeling of the Risk in Mortality Projections

Page 2/34 Introduction 1 Introduction Motivation Literature Review: Mortality Literature Review: Term Structure Models 2 Factor Analysis of Mortality Forecasts 3 Forward Mortality Factor Models 4 A Non-negative Model Variant 5 Application 6 Conclusion Daniel Bauer Coherent Modeling of the Risk in Mortality Projections

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